Determinants+[Practice+Question].pdf

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IIT - ian’s PACE

IIT – ian’s P A C E

216 - 217 ,2nd floor , Shopper’s point , S. V. Road. Andheri (West) Mumbai – 400058 . Tel: 26245223 / 09

Practice Question

LEVEL –1 Determinants

Question

based on Expansion of Determinants

Q.1 If 1 1 1 1 1 1 1 1 1 a   

= 4, then the value a is -

(A) 1 (B) –1 (C) –2 (D) 0 Q.2 If 2 4 y x = 7 and x y 3 2 = 4, then - (A) x = – 3, y = – 2 5 (B) x = – 2 5 , y = – 3 (C) x = 3, y = 2 5 (C) x = 2 5 , y = 3 Q.3 The value of i 5 i 4 i 3 i 5    is - (A) 12 (B) 17 (C) 14 (D) 24 Q.4 x sec x cot x tan 0 1 0 x tan x sin x sec is equal to - (A) 0 (B) – 1 (C) 1 (D) None of these Q.5 The value of y x 1  1 y 5 1 x 3 0 0 1 3 3 _{is -} (A) x + y (B) x2_{– xy + y}2 (C) x2_{+ xy + y}2 _{(D) x}3_{– y}3 Question based on Minors &

Cofactor and their properties

Q.6 The cofactors of 1, –2, –3 and 4 in

4 3 2 1   are- (A) 4, 3, 2, 1 (B) –4, 3, 2, –1 (C) 4, –3, –2, 1 (D) –4, –3, –2, –1

Q.7 The minors of the elements of the first row in the determinant 2 1 1 3 2 4 4 1 2   are- (A) 2, 7, 11 (B) 7, 11, 2 (C) 11, 2, 7 (D) 7, 2, 11 Q.8 If  = 3 3 3 2 2 2 1 1 1 c b a c b a c b a and A₂, B₂, C₂ are

respectively cofactors of a₂, b₂, c₂ then a₁A₂ + b₁B₂ + c₁C₂ is equal to-

(A) –  (B) 0

(C)  (D) None of these

Q.9 If A = (a_ij) is a 4 × 4 matrix and c_ij is the co-factor of the element a_ij in Det (A), then the expression a₁₁c₁₁+ a₁₂c₁₂+ a₁₃c₁₃ + a₁₄c₁₄ equals-

(A) 0 (B) – 1

(C) 1 (D) Det. (A)

Q.10 If cofactor of 2x in the determinant

0 x 1 x 1 x x 2 1 2 1 x   

is zero, then x equals to-

(A) 0 (B) 2

(C) 1 (D) –1

Question

based on Some basic properties

Q.11 The value of the determinant

3 3 3 2 2 2 1 1 1 b ma a b ma a b ma a is - (A) 0 (B) ma₁a₂a₃ (C) ma₁b₂a₂ (D) mb₁b₂b₃

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IIT - ian’s PACE Q.12 If  = b a c a c b 0 0 a , then b a pc a c pb 0 0 a p2 is equal to- (A) p (B) p2_ (C) p3_ _{(D) 2p}

ab 1 c / 1 ca 1 b / 1 bc 1 a / 1 is equal to

(A) abc (B) 1/abc

(C) 0 (D) None of these

Q.14 If each row of a determinant of third order of value  is multiplied by 3, then the value of new determinant is -

(A)  (B) 27  (C) 21  (D) 54 

Q.15 The sum of infinite series

4 6 2 1 + 4 2 2 2 / 1 + 4 3 / 2 2 4 / 1 + ... is - (A) –10 (B) 0 (C) 10 (D)  Q.16 The value of z nz mc c y ny mb b x nx ma a    is- (A) a + b + c (B) x + y + z (C) m(a + b + c) + n(x + y + z) (D) 0 Q.17 The value of c 3 b 6 a 10 b 3 a 6 a 3 c 2 b 3 a 4 b 2 a 3 a 2 c b a b a a          is equal to - (A) a3 _{(B) b}3 (C) c3 _{(D) a}3_{+ b}3_{+ c}3

1 c k kc 1 b k kb 1 a k ka 2 2 2 2 2 2    is - (A) k (a + b) (b + c) (c + a) (B) k abc (a2_{+ b}2_{+ c}2₎ (C) k (a – b) (b – c) ( c – a) (D) k (a + b – c) (b + c – a) (c + a – b) Q.19 If D_r= 2 / ) 1 n 3 ( n z 2 r 3 n y 1 r 2 2 / ) 1 n ( n x r 2     , then



 n 1 r r D is equal to - (A) 6 1 n(n + 1)(2n + 1) (B) 4 1 n2_{(n + 1)}2 (C) 0 (D) None of these Q.20 If x a x a x a x a x a x a x a x a x a          = 0, then value of x are- (A) 0, a (B) 0, – a (C) a, – a (D) 0, 3a

ab ca bc c b a c b a 2 2 2 _{is -} (A) abc (a – b) (b – c) (c – a) (B) (a – b) (b – c) (c – a) (a + b + c) (C) (a – b) (b – c) (c – a) (ab + bc + ca) (D) None of these Q.22 If 2 2 2 2 2 2 2 2 2 ) 1 c ( ) 1 b ( ) 1 a ( ) 1 c ( ) 1 b ( ) 1 a ( c b a       = k 1 1 1 c b a c b a2 2 2 ,

then k is equal to-

(A) 1 (B) 2 (C) 4 (D) 0 Q.23 The value of 3 3 3 c b a c b a c b a c b a    is- (A) (a – b) (b – c) (c – a) (B) abc (a – b) (b – c) (c – a) (C) – (a + b + c)2_{(a – b) (b – c) (c – a)} (D) None of these

Q.24 If x is real number such that

            x 4 x 3 x x 3 x 2 x x 2 x 1 x = 0 then  are in (A) A.P. (B) G.P. (C) H.P. (D) None of these

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IIT - ian’s PACE Q.25 The determinant ab c ca b bc a c b a 1 1 1 2 2 2    is equal to - (A) 0 (B) 1 (C) – 1 (D) None of these Q.26 2 2 m 2 1 m 2 m 1 2 m 1 1 m 1 m C C C C C C 1 1 1     ₌ (A) m(m + 1) (B) m(m – 1) (C) 1 (D) 0

Q.27 Find the value of x in the equation

2 x 5 x 2 1 5 2 1 20 4 1  = 0 (A) –1, 2 (B) –1, 0 (C) 2, 0 (D) 1, 2

Q.28 If a, b, c are in A.P., then the value of

c 2 x 5 x 4 x b 2 x 4 x 3 x a 2 x 3 x 2 x          equals - (A) 1 (B) 0 (C) 2a (D) 2x Q.29 i 1 i 1 i i 1 i i 1 i i 1 i 1       (where i = 1 ) equals - (A) 7 + 4i (B) 7 – 4i (C) 4 + 7i (D) 4 – 7i Q.30 = c 6 b 9 a 11 b 6 a 9 a 6 c 3 b 4 a 5 b 3 a 4 a 3 c b a b a a          where a = i,

b =  c = _{then is equal to-}

(A) i (B) – 2

(C)   (D) – i

2 2 2 2 2 2 2 2 2 ) 2 x ( ) 1 x ( x ) 1 x ( x ) 1 x ( x ) 1 x ( ) 2 x (       is- (A) 0 (B) 8x2 _{(C) 8} _{(D) –8} Q.32 7591 7581 7589 7579 = (A) 20 (B) – 2 (C) – 20 (D) 4 Q.33 If x 6 3 3 3 x 3 6 3 6 x 3       = 0 then x = (A) 6 (B) 3 (C) 0 (D) None of these Question based on

Symmetric and skew symmetric Determinants Q.34 If A + B + C = , then 0 A tan ) B A ( cos A tan 0 B sin C cos B sin ) C B A ( sin      equals-

(A) 0 (B) 2sinB tanA cosC

(C) 1 (D) None of these

Q.35 The value of an even order skew symmetric determinant is-

(A) 0 (B) perfect square

(C) ±1 (D) None of these

Q.36 The value of an odd order skew symmetric determinant is-

(A) perfect square (B) negative

(C) ±1 (D) 0 Q.37 The value of 0 b c a c c b 0 a b c a b a 0       is- (A) 0 (B) abc (C) (a – b) (b – c) (c – a) (D) None of these Question

based on Crammer's Rule

Q.38 The equations x + 2y + 3z = 1,

2x + y + 3z = 2 and 5x + 5y + 9z = 4 have- (A) unique solution (B) many solutions (C) inconsistent (D) None of these

Q.39 The existence of unique solution of the system

x + y + z = b, 2x + 3y – z = 6, 5x – y + az = 10 depends on-

(A) b only (B) a only (C) a and b (D) neither a nor b

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Q.40 Given the system of equations px + y + z = 1, x + py + z = p, x + y + pz = p2_{, then for what}

value of p does this system have no solution -

(A) –2 (B) –1

(C) 1 (D) 0

Q.41 The value of k for which the set of equations 3x + ky – 2z = 0, x + ky + 3z = 0 and 2x + 3y – 4z = 0 has a non – trivial solution is-

(A) 15 (B) 16

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LEVEL

–

2

Q.1 b a c b b a a c b a c c c b a 2 2 2 2 2 2    is equal to-

(A) abc (B) 2abc

(C) 4abc (D) 0 Q.2 1 4 . 3 4 . 3 5 4 1 3 . 3 3 . 3 4 3 1 2 . 3 2 . 3 3 2 2 3 3 2 3 3 2 3 3       is equal to- (A) 0 (B) 1 (C) 92 (D) None of these Q.3 If c b b a a c b a a c c b a c c b b a          =  b a c a c b c b a then

is equal to-

(A) 1 (B) 2 (C) 3 (D) 4 Q.4 If _= r q p z y x c b a and ₂= r c z p a x q b y then ₁is equal to- (A) 2₂ (B) ₂ (C) –₂ (D) None of these Q.5 The determinant 1 1 1 z y x cz by ax 2 2 2 _{is equal to-} (A) 2 2 2 z y x c b a 1 1 1 (B) xy zx yz z y x c b a (C) 2 2 2 c b a z y x 1 1 1 (D) None of these Q.6 If D_p = 10 25 p 9 35 p 8 15 p 3 2 _{, then} D₁ + D₂+ D₃ + D₄ + D₅is equal to- (A) 0 (B) 25 (C) 625 (D) None of these Q.7 If the determinant                      b a a c c b b a a c c b b a a c c b is expressible as m       c b a c b a c b a , then the value of m is- (A) – 1 (B) 0 (C) 1 (D) 2

Q.8 In a third order determinant each element of the first column consists of sum of two terms, each element of the second column consists of sum of three terms and each element of third column consists of sum of four terms, then it can be decomposed into n determinants, where n has the value-

(A) 1 (B) 9 (C) 16 (D) 24

Q.9 For any ABC, the value of determinant

1 C cot C sin 1 B cot B sin 1 A cot A sin 2 2 2 is- (A) 0 (B) 1

(C) sin A sin B sin C (D) sin A + sin B + sin C

Q.10 If S_r = ) 1 n ( n z nr 2 r 4 ) 3 n 2 ( n y 1 r 6 ) 1 n ( n x r 2 3 3 2 2      then



 n 1 r r

S does not depends on

(A) x (B) y

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Q.11 If a, b, c are non-zero real numbers, then

b a ab b a a c ca a c c b bc c b 2 2 2 2 2 2    is equal to (A) abc (B) a2_b2_c2 (C) ab + bc + ca (D) None of these Q.12 x q p q x p q p x is equal to - (A) (x + p) (x + q) (x – p – q) (B) (x – p) (x – q) (x + p + q) (C) (x – p) (x – q) (x – p – q) (D) (x + p) (x + q) (x + p + q)

Q.13 If x is a positive integer then the value of

determinant )! 4 x ( )! 3 x ( )! 2 x ( )! 3 x ( )! 2 x ( )! 1 x ( )! 2 x ( )! 1 x ( ! x         is- (A) (2x) !. (x + 1) !. (x + 2) !. (x + 3) ! (B) 2 (x) !. (x + 1) !. (x + 2) ! (C) (2x) !. (x + 3) ! (D) None of these Q.14 The determinant                cos sin cos sin cos sin 2 cos ) ( sin ) ( cos is- (A) 0 (B) independent of   (C) independent of 

(D) independent of both and  

Q.15 5 15 3 65 10 5 26 15 5 5 2 3 13    equals- (A) 0 (B) 5 3 ( 6 – 5) (C) 5 3 (5 – 6 ) (D) None of these

Q.16 If [a] denotes the greatest integer less than or equal to a and – 1  x < 0, 0  y < 1, 1  z < 2, then 1 ] z [ ] y [ ] x [ ] z [ 1 ] y [ ] x [ ] z [ ] y [ 1 ] x [    is equal to- (A) [x] (B) [y] (C) [z] (D) None of these

1 C C 1 C C 14 C C 5 5 2 5 4 5 1 5 3 5 0 5 is- (A) 0 (B) – (6!) (C) 80 (D) None of these Q.18 If  = b a c a c b c b a

, then 2 _{is equal to-}

(A) 2 2 2 2 2 2 2 2 2 b ca a bc c ab a bc c ab b ca c ab b ca a bc          (B) 2 2 2 2 2 2 2 2 2 b ca a bc c ab a bc c ab b ca c ab b ca a bc          (C) ca b bc a ab c bc a ab c ca b ab c ca b bc a 2 2 2 2 2 2 2 2 2          (D) None of these Q.19 If ax + by + cz = 1, bx + cy + az = 0 = cx + ay + bz, then x z y y x z z y x a c b b a c c b a is equal to- (A) 0 (B) 1 (C) – 1 (D) 2

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LEVEL

–

3

Q.1 If 3 n 2 n n 3 n 2 n n 3 n 2 n n z z z y y y x x x       = (x–y) (y–z) (z–x) _        z 1 y 1 x 1 , then n = (A) 1 (B) –1 (C) 2 (D) –2

Q.2 If  are the roots of x3 _{+ ax}2 _{+ b = 0,}

then the value of

         is equals to- (A) – a3 _{(B) a}3 _–3b (C) a3 _{(D) a}2 _{– 3b}

Q.3 If A, B and C are the angles of a triangle and

0 C sin C sin B sin B sin A sin A sin C sin 1 B sin 1 A sin 1 1 1 1 2 2 2        ,

then the triangle ABC is- (A) isosceles

(B) equilateral

(C) right angled isosceles (D) none of these

1 ) sin( ) cos( 1 cos sin 1 sin cos             is- (A) independent of  (B) independent of  (C) independent of  and  (D) none of these Q.5 Let 1) 4(8 ) 7(8 c 1) 2(4 ) 3(4 b 1 2 2 a D 16 r 16 r 16 r r     , then the value of 16 _r 1 rΣ D is equals to- (A) 0 (B) a + b + c (C) ab + bc + ca (D) none of these

Q.6 The values  and  for which the system of equations x + y + z = 6, x + 2y + 3z = 10 and x + 2y + z = have unique solution are- (A)   3,  R (B)   3,  10 (C)   3,  10 (D)   3,  10

Q.7 The system of linear equations x + y + z = 2, 2x + y – z = 3, 3x + 2y + kz = 4 has a unique solution if- (A) k  (B) –1 < k > 1 (C) –2 < k < 2 (D) k = 0 Q.8 If 1 x 2 1 x 2 3 x 2 x 3 x 3 x 3 1 x 3 x 2 2 x 1 x x x 2 2 2           = Ax – 12,

then the value of A is-

(A) 12 (B) 24 (C) –12 (D) –24 Q.9 The value of 451 450 449 447 446 445 443 442 441 is- (A) 441 × 446 × 451 (B) 0 (C) –1 (D) 1 Q.10 If f(x) = 3 2 1 500 x 4 50 x 2 5 x 81 x 3 18 x 2 3 x 3 2 3 2       then f(1).f(3) + f(3).f(5) + f(5).f(1) is equal to- (A) f(1) (B) f(3) (C) f(1) + f(3) (D) f(1) + f(5)

Q.11 If the system of equations,

x + 2y – 3z =1, (k + 3)z = 3,(2k + 1)x + z = 0 is inconsistent, then the value of k is-

(A) –3 (B) 1/2 (C) 0 (D) 2 Q.12 In a ABC, if 0 c b 1 a c 1 b a 1  then

sin2 _{A + sin}2 _{B + sin}2 _{C is equal to-}

(A) 9/4 (B) 4/9

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Q.13 The equation

x + 2y + 3z = 1, 2x + y + 3z = 2, 5x + 5y + 9z = 4 have-

(A) unique solution

(B) infinitely many solutions (C) inconsistent (D) None of these Q.14 x ) 2 n ( sin x ) 1 n ( sin ) nx ( sin x ) 2 n ( cos x ) 1 n ( cos ) nx ( cos 1 1 1     is not dependent- (A) on x (B) on n

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ANSWER KEY

LEVEL- 1

Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Ans. D B C C C A B B D C A B C B A D A C C D C Q.No. 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 Ans. C C A A C A B C A D C C A B D A A B A D

LEVEL- 2

Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Ans. C A B B B D B D A D D B B B B C D A B

LEVEL- 3

Q.No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Ans. B C A A A A A B B B A A A B